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Mirrors > Home > MPE Home > Th. List > cnncvsmulassdemo | Structured version Visualization version GIF version |
Description: Derive the associative law for complex number multiplication mulass 11226 interpreted as scalar multiplication to demonstrate the use of the properties of a normed subcomplex vector space for the complex numbers. (Contributed by AV, 9-Oct-2021.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
cnncvsmulassdemo | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2728 | . . . 4 ⊢ (ringLMod‘ℂfld) = (ringLMod‘ℂfld) | |
2 | 1 | cncvs 25071 | . . 3 ⊢ (ringLMod‘ℂfld) ∈ ℂVec |
3 | id 22 | . . . 4 ⊢ ((ringLMod‘ℂfld) ∈ ℂVec → (ringLMod‘ℂfld) ∈ ℂVec) | |
4 | 3 | cvsclm 25052 | . . 3 ⊢ ((ringLMod‘ℂfld) ∈ ℂVec → (ringLMod‘ℂfld) ∈ ℂMod) |
5 | 2, 4 | ax-mp 5 | . 2 ⊢ (ringLMod‘ℂfld) ∈ ℂMod |
6 | 1 | cnrbas 25068 | . . . 4 ⊢ (Base‘(ringLMod‘ℂfld)) = ℂ |
7 | 6 | eqcomi 2737 | . . 3 ⊢ ℂ = (Base‘(ringLMod‘ℂfld)) |
8 | cnfldex 21281 | . . . 4 ⊢ ℂfld ∈ V | |
9 | rlmsca 21090 | . . . 4 ⊢ (ℂfld ∈ V → ℂfld = (Scalar‘(ringLMod‘ℂfld))) | |
10 | 8, 9 | ax-mp 5 | . . 3 ⊢ ℂfld = (Scalar‘(ringLMod‘ℂfld)) |
11 | cnfldmul 21286 | . . . 4 ⊢ · = (.r‘ℂfld) | |
12 | rlmvsca 21092 | . . . 4 ⊢ (.r‘ℂfld) = ( ·𝑠 ‘(ringLMod‘ℂfld)) | |
13 | 11, 12 | eqtri 2756 | . . 3 ⊢ · = ( ·𝑠 ‘(ringLMod‘ℂfld)) |
14 | cnfldbas 21282 | . . . . 5 ⊢ ℂ = (Base‘ℂfld) | |
15 | 14 | eqcomi 2737 | . . . 4 ⊢ (Base‘ℂfld) = ℂ |
16 | 15 | eqcomi 2737 | . . 3 ⊢ ℂ = (Base‘ℂfld) |
17 | 7, 10, 13, 16 | clmvsass 25015 | . 2 ⊢ (((ringLMod‘ℂfld) ∈ ℂMod ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))) |
18 | 5, 17 | mpan 689 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 Vcvv 3471 ‘cfv 6548 (class class class)co 7420 ℂcc 11136 · cmul 11143 Basecbs 17179 .rcmulr 17233 Scalarcsca 17235 ·𝑠 cvsca 17236 ringLModcrglmod 21056 ℂfldccnfld 21278 ℂModcclm 24988 ℂVecccvs 25049 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-addf 11217 ax-mulf 11218 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-tpos 8231 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-er 8724 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-fz 13517 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-starv 17247 df-sca 17248 df-vsca 17249 df-ip 17250 df-tset 17251 df-ple 17252 df-ds 17254 df-unif 17255 df-0g 17422 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-grp 18892 df-minusg 18893 df-subg 19077 df-cmn 19736 df-abl 19737 df-mgp 20074 df-rng 20092 df-ur 20121 df-ring 20174 df-cring 20175 df-oppr 20272 df-dvdsr 20295 df-unit 20296 df-invr 20326 df-dvr 20339 df-subrng 20482 df-subrg 20507 df-drng 20625 df-lmod 20744 df-lvec 20987 df-sra 21057 df-rgmod 21058 df-cnfld 21279 df-clm 24989 df-cvs 25050 |
This theorem is referenced by: (None) |
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